Two metal wires of identical dimensions are connected in series. If ${\sigma }_1 and {\sigma }_2$  are the conductivities of the metal wires respectively, the effective conductivity of the combination is

detailed solution

Correct option is C

As both metal wires are of identical dimensions, so their length and area of cross-section will be same. Let them be I and A respectively. Then The resistance of the first wire isR1=lσ1A                                ....(i)and that of the second wire is R2=lσ2A                       ...(ii)As they are connected in series, so their effective resistance isRs=R1+R2=lσ1A+lσ2A  ( using (i) and  (ii))=lA1σ1+1σ2                  . . .(iii)If σeff is the effective conductivity of the combination, thenRs=2lσeff A                        . . . .(iv)Equating eqns. (iii) and (iv), we get 2lσeffA=lA1σ1+1σ22σeff =σ2+σ1σ1σ2⇒σeff =2σ1σ2σ1+σ2